If A and B are two events such that
$P(A) = \frac{1}{2},P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{1}{4}$, then find

(i) $P(A/B)$

(ii) $P(B/A)$

(iii) $P\left( {{A^\prime }/B} \right)$

(iv) $P\left( {{A^\prime }/{B^\prime }} \right)$

Here, $P(A) = \frac{1}{2},P(B) = \frac{1}{3}$

and $P(A \cap B) = \frac{1}{4}$

(i) $P(A/B) = \frac{{P(A \cap B)}}{{P(B)}} = \frac{{1/4}}{{1/3}} = \frac{3}{4}$

(ii) $P(B/A) = \frac{{P(A \cap B)}}{{P(A)}} = \frac{{1/4}}{{1/2}} = \frac{1}{2}$

(iii) $P\left( {{A^\prime }/B} \right) = 1 - P(A/B) = 1 - \frac{3}{4} = \frac{1}{4}$

or $P\left( {{A^\prime }/B} \right) = \frac{{P\left( {{A^\prime } \cap B} \right)}}{{P(B)}} = \frac{{P(B) - P(A \cap B)}}{{P(B)}} = \frac{{\frac{1}{3} - \frac{1}{4}}}{{\frac{1}{3}}} = \frac{{\frac{1}{{12}}}}{{\frac{1}{3}}} = \frac{1}{4}$

(iv) $P\left( {{A^\prime }/{B^\prime }} \right) = \frac{{P\left( {{A^\prime } \cap {B^\prime }} \right)}}{{P\left( {{B^\prime }} \right)}} = \frac{{1 - P(A \cup B)}}{{1 - P(B)}} = \frac{{1 - [P(A) + P(B) - P(A \cap B)]}}{{1 - P(B)}}$

$= \frac{{1 - \left[ {\frac{1}{2} + \frac{1}{3} - \frac{1}{4}} \right]}}{{1 - \frac{1}{3}}} = \frac{{1 - \left( {\frac{5}{6} - \frac{1}{4}} \right)}}{{\frac{2}{3}}}$

$= \frac{{1 - 14/24}}{{2/3}} = \frac{{10/24}}{{2/3}} = \frac{{30}}{{48}} = \frac{5}{8}$